Building complex functions out of real parts

This article discusses how to compute complex-valued functions like sine and cosine using only real functions, referencing Henry G. Baker's equations. The author uses ChatGPT to generate Python code from LaTeX to verify these equations numerically, catching typos and confirming branch cut consistency with NumPy. The post is part of a broader thread on Markov's equation, complex identities, and the definition of inverse elementary functions via logarithms. It delves into advanced topics like branch cuts, 'couth' and 'uncouth' function pairs, and reflects on deeper questions often glossed over in introductory calculus, such as the precise meaning of functions like arccos.